Post on 14-Apr-2022
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Class8:Tensors
Inthisclasswewillexplorehowgeneralco-ordinatetransformationsmaybedescribedbyatensorcalculususingindexnotation,leadingto
ageneralizednotionofcurvature
Class8:Tensors
Attheendofthissessionyoushouldbeableto…
• … understandhowtheLorentztransformationsmaybereplacedbygeneralco-ordinatetransformations
• ...describewhysuchtransformationsarefundamentaltoformulatingthelawsofphysics
• … applyindexnotationtomanipulategeneraltensorobjects,suchasbyraisingorloweringanindex
• … describehowthenotionofparallel-transportinacurvedspaceleadstothegeneralizedRiemanncurvaturetensor
Thelawsofphysics
• AfundamentalideaofRelativityisallreferenceframesareequallysuitablefortheformulationofthelawsofphysics
• Areferenceframeisaspace-timeobservingsystem,suchastheEarth’sframe,orafreely-fallingframe,oraninertialframeinSR
Thelawsofphysics
• Physicsdoesnotdependonourchoiceofco-ordinateframe
• Anequationrepresentingaphysicallawinco-ordinateframe𝑥,suchas𝐴# = 𝐵#,musttransformtoadifferentframe𝑥′suchthat𝐴′# = 𝐵′#
• Weneedsomepowerfulmathematicstoensurethatthiswillhappen– thisisthemathematicsoftensorcalculus
https://comic.hmp.is.it/comic/tensor-calculus/
SpecialRelativityrecap
• InSpecialRelativityweintroducedtheideaofa4-vector,agroupoffourquantities𝐴# whosevaluesin2inertialframesarerelatedbytheLorentztransformations:
• Itwasconvenientforustodefinea“down”4-vector𝐴#:
• Thisisbecause𝐴#𝐴# isaninvariant
𝐴′# = 𝐿#(𝐴(
𝐴# = 𝜂#(𝐴(
𝐿#( =
𝛾 −𝑣𝛾/𝑐−𝑣𝛾/𝑐 𝛾
0 00 0
0 00 0
1 00 1
𝜂#( = 𝜂#( =−1 00 1
0 00 0
0 00 0
1 00 1
𝐴# = 𝜂#(𝐴(
Generaltransformations
• TheLorentztransformationbetweeninertialframesisaspecialcase– wemustdevelopmathematicstodescribeanarbitrarytransformationbetween2co-ordinateframes (e.g.theEarth’sframe,andafreely-fallingframe)
• Aco-ordinatetransformationprovidesrelationsforsome𝑥′co-ordinatesintermsof𝑥 co-ordinates,𝑥2 = 𝑓(𝑥)
https://math.stackexchange.com/questions/1228106/how-can-i-transform-coordinate-systems-based-on-quaternion-data
Generaltransformations
• Westartbytransformingsimpledifferentialsandgradients
usingthechainrule:𝑑𝑥′# = 789:
78;𝑑𝑥( and 7<
789:= 78;
789: 7<78;
• Usingthistemplate…
• A“generalup4-vector”𝐴# isanarraywhosevaluesinthe2
framesarerelatedby:𝑨′𝝁 = 𝝏𝒙9𝝁
𝝏𝒙𝝂𝑨𝝂 (theLorentz
transformationisaspecialcaseofthiswith𝑥′# = 𝐿#(𝑥()
• A“generaldown4-vector”𝐴# isanarraywhosevaluesinthe
2framesarerelatedby:𝑨′𝝁 =𝝏𝒙𝝂
𝝏𝒙9𝝁𝑨𝝂
Generaltransformations
• Thegeneraltransformationofan“up”indextoa“down”indexusesthespace-timemetric:
• 𝒈𝝁𝝂 istheinversematrixof𝒈𝝁𝝂,sinceapplyingbothoftheseoperationsinturnto𝐴# mustrestoretheoriginalquantity
• Inaninertialorfreely-fallingframe,𝑔#( = 𝜂#(,andwerecoverthepreviousrulesforraising/loweringanindex
𝐴# = 𝑔#(𝐴( 𝐴# = 𝑔#(𝐴(
Tensors
• Somephysicalquantitiesaregroupedintolargerstructures
• Moregenerally,atensor 𝐴#( transformsbetween2framesas:
• Weraiseandlowerindicesusingthemetric,forexample:
• Wecangeneralizetheserelationstohigherdimensions
𝐴′#( =𝜕𝑥2#
𝜕𝑥E 𝜕𝑥2(
𝜕𝑥F𝐴EF 𝐴′#( =
𝜕𝑥E
𝜕𝑥2# 𝜕𝑥F
𝜕𝑥2( 𝐴EF
𝐴F( = 𝑔F#𝐴#(
𝐴#F = 𝑔F(𝐴#(
𝐴EF = 𝑔E#𝑔F(𝐴#(
Tensors
Withthismathematicalapparatusinhandwecanderiveanumberofusefulrelationsoftensorcalculus:
• If𝐴# = 𝐵# then,inanyotherframe,𝐴′# = 𝐵′#
• 𝐴#𝐵( isatensor𝐶#(
• 𝐴#𝐵# isascalarinvariantinallframes
• 𝐶#(𝐴( isa4-vector𝐷#
• Wecanre-arrangesummedindices,e.g.𝐴#𝐵# = 𝐴#𝐵#
Wehavealreadymetsometensorsinthecourse,suchas𝑔#(and𝑇#(.Weareabouttomeetsomemore!
Generaldescriptionofcurvature
• Howcanwedescribethecurvatureofaregionofspace?
• Onthesurfaceofasphere,carryanarrowfromtheEquatortothePoleandbackonapath𝐴 → 𝑁 → 𝐵 → 𝐴 shownbelow
• Supposeweparallel-transport thearrow,meaningthatitscomponentsareunchangedinalocalCartesiansystem
https://commons.wikimedia.org/wiki/File:Parallel_transport.png
Vectorsparallel-transportedaroundaclosedpathonacurvedsurfacearerotated!
Generaldescriptionofcurvature
• Supposeyouareatapoint𝑥# inspace-time
• Travelindirection𝑖 untilyourco-ordinate𝑥M isincreasedbyasmallamount𝑑𝑥M,withoutchangingtheotherco-ordinates
• Nowtravelindirection𝑗 untilco-ordinate𝑥O isincreasedby𝑑𝑥O,withoutchangingtheotherco-ordinates
• Nowmovebackwardsby−𝑑𝑥M
• Finally,movebackwardsby−𝑑𝑥O
• Youare,ofcourse,backat𝑥#! 𝑑𝑥M
𝑑𝑥O
• Nowtravelthesamerouteagain,parallel-transportingavector𝐴P whichinitiallypointsindirection𝑘
• Vectorsparallel-transportedaroundaclosedpathinacurvedsurfacearerotated– letthechangeineachcomponentbe𝑑𝐴R
• ThisthoughtexperimentallowsustodefinetheRiemanntensor𝑹𝝀𝝁𝝂𝜿 ,whichprovidesageneralmeasureofcurvature:
• TheRiemanntensormaybeexpressedintermsoftheChristoffel symbols: 𝑅F#(
E = 𝜕#ΓF(E − 𝜕(ΓF#
E + Γ#YE ΓF(Y − Γ(YE ΓF#
Y
𝑑𝐴R = 𝑅PMOR 𝐴P𝑑𝑥M𝑑𝑥O
TheRiemanntensor
TheRiemanntensor
• Inan𝑁-dimensionalspace,wehave𝑁Z
possibleloops,and𝑁 finalcomponentsof𝑁 initialvectors,i.e.𝑁[ componentsateachpoint,= 256 for𝑁 = 4!!
• Itisnotthatbad,owingtosymmetries(e.g.,goingbackwardsroundtheloop).Actually,thenumberofindependentcomponentsis𝑵𝟐(𝑵𝟐 − 𝟏)/𝟏𝟐
• Sothecurvatureateachpointisdescribedby1numberwhen𝑁 = 2(ona2Dcurvedsurface),and20numberswhen𝑁 = 4